All Questions
Tagged with differentiation electromagnetism
109 questions
3
votes
1
answer
67
views
"Deriving" the covariant derivative
Suppose we are working in scalar QED with Lagrangian
$$\mathscr{L} = (D_\mu \phi)(D^\mu \phi)^* - \frac{1}{4}F_{\mu\nu}F^{\mu\nu}.$$
I now want to find the form of the covariant derivative $D_\mu$ ...
- 283
0
votes
1
answer
75
views
Derivative wrt retarded time
I am confused by the following statement in footnote of Griffiths 4th edition (page 446):
$$\frac{\partial }{\partial t_r} = \frac{\partial }{\partial t},$$ where $$t_r=t - \frac{\mathscr{r}}{c}$$ ...
5
votes
4
answers
386
views
Vector triple product with $\nabla$ operator
I came across the following expression in several books (especially in plasma physics literature while deriving the magnetic pressure):
$$(\mathbf{\nabla} \times \mathbf{B})\times \mathbf{B} = \left(\...
- 61
1
vote
1
answer
71
views
Meaning of colon symbol $:$ in optics
When I was reading some early days nonlinear optics paper/textbooks (particularly between 1960-1985), I often see expressions such as:
$\chi^{(2)}:\textbf{E}\textbf{E}$
or
$\nabla\textbf{E}:\partial \...
- 173
0
votes
3
answers
133
views
Field strength tensor written as commutator of covariant derivatives in QED
I am currently trying to understand the derivation of the relation
$$
\begin{equation}
F_{\mu\nu} = \frac{1}{iq}[D_{\mu}, D_{\nu}]\tag{1}\label{eq1}
\end{equation}
$$
in QED and I have trouble with ...
- 3
0
votes
2
answers
93
views
Why is $(\partial_\mu F_{\alpha\beta})F^{\alpha\beta}=F_{\alpha\beta}\partial_\mu(F^{\alpha\beta})$?
I'm trying to prove that the divergence of the energy-momentum-tensor is zero by expressing it in terms of the field strength tensor: $\partial_\mu T^{\mu\nu}=0$.
In doing this, letting the derivative ...
- 33
1
vote
1
answer
67
views
How do you differentiate $F^{αβ}$ with respect to $g_{μν}$?
I want to experiment with this relation (from Dirac's "General Theory of Relativity"):
$$T^{μν} = -\left(2 \frac{∂L}{∂g_{μν}} + g^{μν} L \right)$$
using the electromagnetic Lagrangian $L = -(...
- 467
3
votes
1
answer
163
views
Laplace-Beltrami operator for a vector field
For a scalar field $\varphi$, the "wave" operator is defined as follows:
$$\Box \varphi \equiv g^{ab}\nabla_a\nabla_b~\varphi = \frac{1}{\sqrt{|g|}}\partial_a\left\{\sqrt{|g|}~g^{ab}~\...
- 603
0
votes
1
answer
111
views
Solving divergence and curl equations numerically
I've recently come to learn about Jefimenko's general solution for Maxwell's equations as well as the FDTD method in electromagnetic optics, and that has got me thinking whether I myself can solve ...
- 1,880
3
votes
1
answer
310
views
What's the physical meaning of Curl of Curl of a Vector Field?
The curl of curl of a vector field is, $$\nabla \times (\nabla \times \mathbf{A}) = \nabla (\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A}$$
Now, curl means how much a vector field rotates ...
- 121
1
vote
1
answer
62
views
Deriving the Curl of the Magnetic Field, Role of the Nabla Operator
We know that the magnetic field can be written in the following way:
$$\nabla_{\vec r }\times\vec B(\vec r) = \frac 1 c \nabla_{\vec r}\times\int d^3\vec r_q\ \vec j(\vec r_q)\times \frac {\vec r-\vec ...
- 193
0
votes
3
answers
363
views
How to derive $i=I_0 \sin(wt)$ in alternating current? [closed]
Our teacher taught us today that instantaneous value of current in Alternating Current is
$$i=I_0 \sin(wt)$$
Where $I_0$ is the amplitude and $wt$ is the angular speed times time. Now, she didn't ...
- 1
0
votes
0
answers
117
views
Are eigenvalues of slashed covariant derivative real?
I am trying to demonstrate that the slashed covariant derivative
$$
\gamma^\mu D_\mu = \gamma^\mu(\partial_\mu -iA_\mu)
$$
has real eigenvalues:
$$
\gamma^\mu D_\mu \varphi_m(x)=\lambda_m \varphi_m(x)...
- 161
1
vote
0
answers
31
views
Problem in calculation of spherically symmetric Laplacian in electrodynamics
I have come across the following operation in two electrodynamics textbooks, which I find problematic: When evaluating an integral over a Laplacian in a spherically symmetric function, the radial term ...
- 934
3
votes
3
answers
116
views
Finding the vector potential
$$\nabla\times\mathbf{B}=\nabla\times\left(\nabla\times\mathbf{A}\right)=\nabla\left(\nabla\cdot\mathbf{A}\right)-\nabla^2\mathbf{A}=\mu_0\mathbf{J}\tag{5.62}$$
Whenever I try to work this out and ...
- 220
0
votes
1
answer
70
views
How to understand the derivatives in wave equation?
I am looking at the derivation of the wave equation, but I am stuck on the math. Specifically, in the following:
How do they get the equivalence between $\frac{\partial}{\partial z} (\frac{dg}{du}) = ...
- 149
3
votes
1
answer
92
views
Bianchi identity in EMT [closed]
$ ∇_a∇_b F_{ab} = 0 $ ($F_{ab}$ Faraday tensor in EMT.)
proof is given by
"To see this, assume a Minkowski spacetime for simplicity and adopt
Cartesian coordinates, so that the covariant ...
- 81
0
votes
1
answer
183
views
Why the divernce of this magnetic field is not zero?
I am working on a project on which I need to calculate the geomagnetic field in different coordinates. When I use the conventional form of the dipole field in spherical coordinates:
$$\vec{B}_{r,\phi}=...
3
votes
3
answers
579
views
How does Kirchhoff's voltage law relate to the spatial derivative of voltage?
I'm reading this libretexts article on the basics of transmission line theory. In it, they include this circuit diagram as a model of a uniform transmission line:
They then say that applying ...
- 2,373
3
votes
4
answers
638
views
Derivation of covariant derivative
I'm currently doing Introductory QFT and was confused about the origin of the additional terms in the covariant derivate. My understanding is as follows:
If we begin with the Dirac Lagrangian ...
- 100
1
vote
0
answers
187
views
Lienard-Wiechert Potential derivation in Wald's "Advanced Classical Electromagnetism" [closed]
I want to follow the Lienard-Wiechert potential derivation in Robert Wald's E-M book, page 179. I do not understand $dX(t_\text{ret})/dt$ on the right side. I assume the chain rule is applied and $x'^...
- 149
4
votes
1
answer
230
views
Is there a quick way to calculate the derivative of a quantity that uses Einstein's summation convention?
Consider $F_{\mu\nu}=\partial_{\mu}A_\nu-\partial_\nu A_\mu$, I am trying to understand how to fast calculate $$\frac{\partial(F_{\mu\nu}F^{\mu\nu})}{\partial (\partial_\alpha A_\beta)}$$
without ...
- 862
1
vote
1
answer
258
views
Derive interaction lagrangian for KG equation in QED
The free-field KG lagrangian density for complex scalar field is given as $$\hat{\mathcal{L}}_{\text{KG}}=(\partial_\mu\hat{\phi}^\dagger)(\partial^\mu\hat\phi)-m^2\hat{\phi}^\dagger\hat{\phi}$$
By ...
- 862
1
vote
1
answer
246
views
Four-vector differentiation (E-M Euler-Lagrange eq.)
$$\partial_{\mu} \frac{\partial(\partial_{\alpha}A_{\alpha})^2}{\partial(\partial_{\mu}A_{\nu})} = \partial_{\mu}\left[2(\partial_{\alpha}A_{\alpha})\frac{\partial(\partial_{\beta}A_{\gamma})}{\...
- 149
1
vote
3
answers
1k
views
Field strength tensor in spherical coordinates
I'm getting confused by the change of coordinates when calculating the electromagnetic tensor in spherical coordinates. In particular I know that in cartesian coordinates:
$$F_{\mu \nu}=\partial_{\mu}...
- 771
1
vote
1
answer
95
views
How to show the equivalence between Lagrangians?
I have a Lagrangian of a form:
$$\mathcal{L}=\frac{1}{2}\left (A_\mu g^{\mu\nu}\partial^2 A_\nu-A_\mu \partial^\mu \partial^\nu A_\nu\right ) $$
And I want to show that it is equivalent to the ...
- 47
0
votes
1
answer
47
views
Divergence applied to vector field, question
Divergence is defined as a scalar valued function:
$$
\left. \operatorname{div} \mathbf{F} \right|_\mathbf{x_0} = \lim_{V \to 0} \frac{1}{|V|} \int\int_{\scriptstyle S(V)}
\mathbf{F} \cdot \mathbf{\...
- 245
0
votes
1
answer
60
views
How can I prove this relation between derivatives? [closed]
Consider coaixialcable with TEM. Nonstatic fields are being considered, i.e situation obeys $\nabla \times \mathbf {E}=-\frac{\partial \mathbf{B} }{\partial t} $
If I let a eletric field be described ...
- 1
0
votes
1
answer
91
views
Poynting Theorem Derivation
I found this equation when I read about Poynting's theorem in Griffith's book.
$$
\textbf{B}\cdot\frac{\partial\textbf{B}}{\partial t}=\frac{1}{2}\frac{\partial}{\partial t}(B^2)
$$
Can anyone please ...
1
vote
3
answers
824
views
Why is emf equal to the rate of change of magnetic flux?
I don't understand how Faraday figured out that the emf induced when a magnet is moved in a coil would be equal to the rate of change of magnetic flux. Yes, I have that formula drilled in my head like ...
- 753
0
votes
3
answers
114
views
What and how are you measuring with $\frac{dy}{dx}$, $dx$, $\mathbf{\nabla} \cdot$ and $\mathbf{\nabla} \times$?
In geometry, the gap between the mathematical model, be it axiomatic or algebraic, and the measurable real world is often non-existent. We have the intuition going from the mathematical model to world ...
- 229
1
vote
2
answers
70
views
Mandl & Shaw QFT chapter 1 question [closed]
Page 3 of Mandl & Shaw claims that, given a vector $\pmb{A}(\pmb{x},t)=\pmb{A}_{0}e^{i(\pmb{k}\pmb{\cdot} \pmb{x} - \omega t)}$, $\pmb{\nabla} \pmb{\cdot} \pmb{A} = 0$ (eq. 1.6) implies $\pmb{k} \...
- 21
-1
votes
1
answer
99
views
What does $\frac{\partial}{\partial t}\delta(\mathbf{r}-\mathbf{r}_k)$ equal to?
How do we get the gradient in the RHS of (2.15), where $\mathbf{r}_k(t)$ is the position of the moving particle? This is from page 32 in Zangwill's electrodynamics textbook:
Let $N$ point charges $...
1
vote
1
answer
432
views
Gauge covariant derivative of an adjoint action: $\psi(x) \to g \psi(x) g^{-1}$, instead of a left action $\psi(x)\to e^{iq\theta(x)} \psi(x)$
In the case where the transformation on $\psi$ is applied from the left:
$$
\psi(x)\to e^{-iq\theta(x)}\psi(x).
$$
The gauge covariant derivative is
$$
D_\mu = \partial_\mu - iqA_\mu \tag{1}
$$
and ...
- 1,558
2
votes
1
answer
280
views
Factor before Dirac delta in magnetic dipole field formula
I bumped into this formula for the magnetic induction field generated by a dipole, containing Dirac's delta, while studying hyperfine splitting: $$\textbf{B}(\textbf{r}) = \frac{2}{3}\mu_0 \textbf{m}\...
1
vote
1
answer
73
views
Finding the maximum electric field strength above a ring with a hole in the middle
I'm doing a problem (not homework, by the way) which asks for the electric field strength on the axis of symmetry a distance $x$ above the centre of a circular disc, which has uniform surface charge ...
- 221
1
vote
1
answer
140
views
Intuitive Definition of Curl and Stokes' Theorem
I am asking this question on SE Physics because it's about an intuitive derivation, not a rigorously made proof of a theorem.
Reading Mathematical Methods for Physics and Engineering by Riley, et al. (...
- 485
4
votes
2
answers
59
views
Where does one more '$\rm m$' come from in the units?
$$\nabla \times A = B$$
$A$ is vector magnetic potential, $\mathrm{Wb/m}$
$B$ is magnetic field intensity, $\mathrm{Wb/m^2}$
Where does one more m come from for $B$? Is that from the gradient operator ...
- 285
0
votes
1
answer
132
views
Vector calculus in Electromagnetism [closed]
I found a problem which had $$\partial_i (A_j \vec{G})= (\vec{\nabla} .\vec{ A} )\vec{G}+ (\vec{A}.\nabla) \vec{G} $$ but my problem is what does $$\partial_i (A_j \vec{B})$$ even mean? it doesn't ...
- 63
2
votes
1
answer
45
views
Why does linear media produce a $1/2$ factor when using partial derivatives?
Oftentimes in Jackson's text there are certain remarks made about linear media (see pg. 226, ed. 2, for example) and there is often a simplification of partial differentials or variations made. For ...
- 134
2
votes
1
answer
373
views
Gauge covariant derivative, - how do I get the field?
Suppose I wish to create a gauge covariant derivative from
$$
\psi(x)\to e^{ia(x)}\psi(x)
$$
I first note that the usual derivate is not covariant:
$$
\partial_x(e^{ia(x)}\psi(x))=i(\partial _xa(x))e^{...
- 1,558
1
vote
1
answer
404
views
Difference between covariant derivatives in general relativity and electromagnetism
There are many similarities between the gauging of a $U(1)$ symmetry to obtain the physics of electromagnetism and the gauging of diffeomorphisms to obtain the physics of general relativity. In ...
-1
votes
1
answer
98
views
How we can prove this vector identity?
I was trying to rederive the formula of the angular momentum of electromagnetic field, and all the steps are clear for me except this one which I took from
"Photons and Atoms: Introduction to ...
- 11
4
votes
1
answer
111
views
What does $\mathbf{A}\cdot\nabla$ mean here?
What does $\mathbf{A}\cdot\nabla$ mean in an expression like $(\mathbf{A}\cdot\nabla)\mathbf B$?
I found this in Griffiths’ Classical Electrodynamics book and cannot figure it out.
1
vote
1
answer
604
views
Visual representation of the curl of the magnetic vector potential!
I know that the electric field (a vector field) is the result of the gradient of the electric potential,which is a scalar field of the type: $\Phi$ : $\mathbb{R}^3 \rightarrow \mathbb{R}$. So the ...
- 1,628
0
votes
0
answers
83
views
Doubt of gauge covariant derivatives: how can I derive it?
In the context of general relativity (GR) it is necessary to introduce the notion of covariant derivatives. From the point of view of a basic introduction, we always start to deal with GR in a highly ...
- 3,159
3
votes
2
answers
739
views
Derivation of curl of magnetic field in Griffiths
Can someone please derive how $$\frac{d}{dx} f(x-x') = -\frac{d}{dx'} f(x-x')~?$$
In Griffiths electrodynamics, this is directly mentioned. I'm really confused, can someone elaborate!
- 453
0
votes
1
answer
1k
views
Commutator of covariant derivative and field $F_{\mu \nu}$
I am working with the covariant derivative and trying to show that the commutator of this derivative
$[D_\mu , D_\nu]$ is proportional to the field $F_{\mu \nu}$. That is, I need the final term to
be ...
user276724
3
votes
1
answer
454
views
Heaviside-Feynman formula derivation
I want to discuss derivation of Feynman-Heaviside formula.
The topic has already been discussed here but I can not put there any question that's why I'm making new post.
Deriving Heaviside-Feynman ...
0
votes
2
answers
609
views
How does a charged particle behave in a vector potential?
I know that a charged particle interacts with a magnetic field through the Lorentz force, thus knowing how it behaves in a given magnetic field.
However, I don't understand how a charged particle (be ...
